Monoidal adjunction

Suppose that $(\mathcal C,\otimes,I)$ and $(\mathcal D,\bullet,J)$ are two monoidal categories. A monoidal adjunction between two lax monoidal functors

(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J)

and

(G,n):(\mathcal D,\bullet,J)\to(\mathcal C,\otimes,I)

is an adjunction $(F,G,\eta,\varepsilon)$ between the underlying functors, such that the natural transformations

\eta:1_{\mathcal C}\Rightarrow G\circ F

and

\varepsilon:F\circ G\Rightarrow 1_{\mathcal D}

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J)

is a lax monoidal functor such that the underlying functor $F:\mathcal C\to\mathcal D$ has a right adjoint $G:\mathcal D\to\mathcal C$ . This adjuction lifts to a monoidal adjuction $(F,m)$ ⊣ $(G,n)$ if and only if the lax monoidal functor $(F,m)$ is strong.

Monoidal adjunction

Lifting adjunctions to monoidal adjunctions

See also